Properties

Label 4-2e2-1.1-c37e2-0-0
Degree $4$
Conductor $4$
Sign $1$
Analytic cond. $300.772$
Root an. cond. $4.16446$
Motivic weight $37$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.24e5·2-s − 5.01e8·3-s + 2.06e11·4-s + 4.18e12·5-s − 2.63e14·6-s − 3.51e15·7-s + 7.20e16·8-s + 1.12e17·9-s + 2.19e18·10-s + 2.56e19·11-s − 1.03e20·12-s − 1.50e20·13-s − 1.84e21·14-s − 2.09e21·15-s + 2.36e22·16-s + 1.65e23·17-s + 5.89e22·18-s + 3.15e23·19-s + 8.62e23·20-s + 1.76e24·21-s + 1.34e25·22-s + 2.34e25·23-s − 3.61e25·24-s − 1.26e26·25-s − 7.88e25·26-s − 2.12e26·27-s − 7.24e26·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.747·3-s + 3/2·4-s + 0.490·5-s − 1.05·6-s − 0.815·7-s + 1.41·8-s + 0.249·9-s + 0.693·10-s + 1.39·11-s − 1.12·12-s − 0.370·13-s − 1.15·14-s − 0.366·15-s + 5/4·16-s + 2.84·17-s + 0.352·18-s + 0.695·19-s + 0.735·20-s + 0.609·21-s + 1.96·22-s + 1.50·23-s − 1.05·24-s − 1.73·25-s − 0.524·26-s − 0.702·27-s − 1.22·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+37/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Analytic conductor: \(300.772\)
Root analytic conductor: \(4.16446\)
Motivic weight: \(37\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4,\ (\ :37/2, 37/2),\ 1)\)

Particular Values

\(L(19)\) \(\approx\) \(6.226870658\)
\(L(\frac12)\) \(\approx\) \(6.226870658\)
\(L(\frac{39}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - p^{18} T )^{2} \)
good3$D_{4}$ \( 1 + 167228936 p T + 21233264503222 p^{8} T^{2} + 167228936 p^{38} T^{3} + p^{74} T^{4} \)
5$D_{4}$ \( 1 - 33460806876 p^{3} T + 73668826362342135734 p^{9} T^{2} - 33460806876 p^{40} T^{3} + p^{74} T^{4} \)
7$D_{4}$ \( 1 + 501811279648208 p T + \)\(70\!\cdots\!98\)\( p^{4} T^{2} + 501811279648208 p^{38} T^{3} + p^{74} T^{4} \)
11$D_{4}$ \( 1 - 25659945722560373304 T + \)\(60\!\cdots\!66\)\( p^{3} T^{2} - 25659945722560373304 p^{37} T^{3} + p^{74} T^{4} \)
13$D_{4}$ \( 1 + \)\(15\!\cdots\!28\)\( T - \)\(38\!\cdots\!02\)\( p^{2} T^{2} + \)\(15\!\cdots\!28\)\( p^{37} T^{3} + p^{74} T^{4} \)
17$D_{4}$ \( 1 - \)\(97\!\cdots\!92\)\( p T + \)\(27\!\cdots\!06\)\( p^{3} T^{2} - \)\(97\!\cdots\!92\)\( p^{38} T^{3} + p^{74} T^{4} \)
19$D_{4}$ \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(95\!\cdots\!62\)\( p T^{2} - \)\(31\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
23$D_{4}$ \( 1 - \)\(44\!\cdots\!48\)\( p^{2} T + \)\(11\!\cdots\!18\)\( p^{2} T^{2} - \)\(44\!\cdots\!48\)\( p^{39} T^{3} + p^{74} T^{4} \)
29$D_{4}$ \( 1 + \)\(53\!\cdots\!80\)\( p T + \)\(30\!\cdots\!98\)\( p^{2} T^{2} + \)\(53\!\cdots\!80\)\( p^{38} T^{3} + p^{74} T^{4} \)
31$D_{4}$ \( 1 - \)\(15\!\cdots\!24\)\( p T + \)\(21\!\cdots\!46\)\( p^{2} T^{2} - \)\(15\!\cdots\!24\)\( p^{38} T^{3} + p^{74} T^{4} \)
37$D_{4}$ \( 1 + \)\(12\!\cdots\!56\)\( T + \)\(21\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!56\)\( p^{37} T^{3} + p^{74} T^{4} \)
41$D_{4}$ \( 1 + \)\(11\!\cdots\!56\)\( T + \)\(10\!\cdots\!46\)\( T^{2} + \)\(11\!\cdots\!56\)\( p^{37} T^{3} + p^{74} T^{4} \)
43$D_{4}$ \( 1 - \)\(10\!\cdots\!52\)\( T + \)\(41\!\cdots\!62\)\( T^{2} - \)\(10\!\cdots\!52\)\( p^{37} T^{3} + p^{74} T^{4} \)
47$D_{4}$ \( 1 - \)\(16\!\cdots\!04\)\( T + \)\(13\!\cdots\!78\)\( T^{2} - \)\(16\!\cdots\!04\)\( p^{37} T^{3} + p^{74} T^{4} \)
53$D_{4}$ \( 1 - \)\(37\!\cdots\!72\)\( T + \)\(21\!\cdots\!22\)\( T^{2} - \)\(37\!\cdots\!72\)\( p^{37} T^{3} + p^{74} T^{4} \)
59$D_{4}$ \( 1 - \)\(12\!\cdots\!60\)\( T + \)\(10\!\cdots\!38\)\( T^{2} - \)\(12\!\cdots\!60\)\( p^{37} T^{3} + p^{74} T^{4} \)
61$D_{4}$ \( 1 + \)\(13\!\cdots\!16\)\( T + \)\(23\!\cdots\!06\)\( T^{2} + \)\(13\!\cdots\!16\)\( p^{37} T^{3} + p^{74} T^{4} \)
67$D_{4}$ \( 1 - \)\(38\!\cdots\!04\)\( T + \)\(31\!\cdots\!58\)\( T^{2} - \)\(38\!\cdots\!04\)\( p^{37} T^{3} + p^{74} T^{4} \)
71$D_{4}$ \( 1 - \)\(14\!\cdots\!24\)\( T + \)\(67\!\cdots\!26\)\( T^{2} - \)\(14\!\cdots\!24\)\( p^{37} T^{3} + p^{74} T^{4} \)
73$D_{4}$ \( 1 - \)\(58\!\cdots\!72\)\( T + \)\(13\!\cdots\!02\)\( T^{2} - \)\(58\!\cdots\!72\)\( p^{37} T^{3} + p^{74} T^{4} \)
79$D_{4}$ \( 1 + \)\(16\!\cdots\!80\)\( T + \)\(39\!\cdots\!18\)\( T^{2} + \)\(16\!\cdots\!80\)\( p^{37} T^{3} + p^{74} T^{4} \)
83$D_{4}$ \( 1 - \)\(65\!\cdots\!12\)\( T + \)\(22\!\cdots\!82\)\( T^{2} - \)\(65\!\cdots\!12\)\( p^{37} T^{3} + p^{74} T^{4} \)
89$D_{4}$ \( 1 - \)\(69\!\cdots\!80\)\( T + \)\(91\!\cdots\!58\)\( T^{2} - \)\(69\!\cdots\!80\)\( p^{37} T^{3} + p^{74} T^{4} \)
97$D_{4}$ \( 1 - \)\(58\!\cdots\!44\)\( T + \)\(69\!\cdots\!58\)\( T^{2} - \)\(58\!\cdots\!44\)\( p^{37} T^{3} + p^{74} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.52918676248945698910236127338, −18.92295359697899073731511353687, −17.14710500777290908657244849836, −16.87358798325217309157824296804, −15.84735804575658006118092166087, −14.72920695550650551831062717891, −14.01021100532241977055645552609, −13.08662161694281413678592112086, −11.93759671991908892679954913749, −11.83403183625011824699565345154, −10.24672736015074998865242365012, −9.519283170510364101715421510297, −7.51659767810903556344756728415, −6.61171213408361419346297653220, −5.69411201142775513597962334611, −5.23412276945900515983514671119, −3.72453168370163521021637470905, −3.24531358055698392923568953908, −1.73529952177444601908514277158, −0.844811819902837591930642767390, 0.844811819902837591930642767390, 1.73529952177444601908514277158, 3.24531358055698392923568953908, 3.72453168370163521021637470905, 5.23412276945900515983514671119, 5.69411201142775513597962334611, 6.61171213408361419346297653220, 7.51659767810903556344756728415, 9.519283170510364101715421510297, 10.24672736015074998865242365012, 11.83403183625011824699565345154, 11.93759671991908892679954913749, 13.08662161694281413678592112086, 14.01021100532241977055645552609, 14.72920695550650551831062717891, 15.84735804575658006118092166087, 16.87358798325217309157824296804, 17.14710500777290908657244849836, 18.92295359697899073731511353687, 19.52918676248945698910236127338

Graph of the $Z$-function along the critical line